Avoiding thermal noise to enter low temperature while not using dissipative element

Question

Let's assume that we want to interact with a circuit at very low temperature \$T2\$ to avoid any source of noise. The voltage interacting with this circuit must have a specific target value \$V0\$. Imagine for instance a waveguide terminated by an LC and we want that the voltage felt by this LC circuit matches \$V0\$ the most closely. But our constraint is that the signals are generated at \$T1 \gg T2\$

One issue will be that the Nyquist noise at \$T2\$ might annoy us, and we want to remove it. One way to avoid it is to put an attenuator at \$T2\$. It will dissipate (almost) all the noise at its input (and thus the noise emitted at \$T1\$). The drewback is that it is dissipative by essence and we thus loose energy.

I wanted to know if it is possible to make a circuit that realizes the same goal (isolating the low temperature stage from noise at high temperature) without making use of any dissipative elements. As a constraint we must avoid any unwanted reflection of signal on the low temperature stage so that if one wave has been sent it will only interact once with the low temperature circuit (no extra reflection will come to annoy us).

I saw that for a two port circuit what I ask is impossible. I call \$S\$ the scattering matrix. Assuming a two port device, calling 1 the port coming from high temperature and 2 the one going to low temperature, a lossless circuit will have to verify:

\$|S_{11}|=|S_{22}|\$,\$|S_{12}|=|S_{21}|\$, \$|S_{11}|=\sqrt{1-|S_{12}|}\$

Here, I need to at least ask: \$S_{22}=0\$ (to avoid those extra reflections). And this gives me: \$|S_{11}|=|S_{22}|=0\$,\$|S_{12}|=|S_{21}|=1\$. Thus basically all the signal (including noise) from high temperature is entirely transmitted which is something I don't want.

Are there way to make this work by considering \$n>2\$ port circuits ? Calling \$2\$ the port associated to the circuit I want to control, assuming that all port but \$2\$ are connected in the end to the temperature \$T1\$ we want to limit thermal noise while being able to control it via the port \$1\$ while using non dissipative elements. I cannot just send a huge signal amplitude in the beginning to have a very high signal over noise ratio because I also must match a specific \$V0\$ at the end of my circuit.

Answer 1

Usually, in these situations what really matters is to avoid attenuation in the return path from the cryogenic device under test (shown at the top of the sketch), since you are trying to detect a signal from it with good SNR. Attenuation in the excitation signal coming from the high temperature is much less of a problem, since the signal source can have whatever power is required. If this is your situation, you can use a cryogenic circulator and attenuator. (The circulator sends a signal entering port 1 to port 2, entering port 2 to port 3, and entering port 3 to port 1, ideally with very little attenuation or reflection.) The excitation signal passes through a cryogenic attenuator before it goes to the circulator, to remove broadband high-temperature thermal noise. Broadband noise coming from the amplifier ends up in the attenuator, rather than in the device under test.